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%{\large{\bf Rate equations for the simulation of cell evolution}}\\
%{\large{\bf February 16, 2011}}
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In the equations, the numbers in the subscripts refer to a ``type'' of mRNA or protein.
Each constant specific to this type also has the same type number. For example,
the transcription rate for protein 1 is written as $k_{txn_1}$.
The superscript refers to the number of post translational modifications
(PTMs) a protein has gone through.
$P^0$ is the basic protein which does not have any PTMs.
The implementation defines the 
maximum number of PTMs in a constant called $PTM\_max$.

\rule{\textwidth}{0.5mm}
{\bf Equation set 1: Initialization of one gene}\\
\rule{\textwidth}{0.5mm}

\begin{equation}
\label{eq:mrna}
\frac{d[mRNA_1]}{dt} =
G_1 \times k_{txn_1} - [mRNA_1] * k_{deg\_mRNA_1}
\end{equation}
where

\begin{equation}
G_1 = \frac {1}{1+(\frac{k_{f_1}}{k_{r_1}})^{Hill_1}}
\end{equation}

$G_1$ refers to the {\em Goodwin term} which is a model derived from the
cooperativity assumption. The constants $k_{f_1}$ and $k_{r_1}$ refer
to the rates of forward transcription and reverse transcription,
respectively. 

In equation (\ref{eq:mrna}), the first  term refers to the mRNAs that
are being transcribed and thus are added to the system. The second term
refers to the mRNAs that are being degraded and thus are taken out
of the system.

\begin{equation}
\label{eq:basicp}
\frac{d[P_1^0]}{dt} = [mRNA_1] \times k_{tsln\_{P_1}} - [P_1] \times k_{deg\_{P_1}}
\end{equation}

In equation (\ref{eq:basicp}), the first  term refers to the proteins that
are being translated and thus are added to the system. The second term
refers to the proteins that are being degraded and thus are taken out
of the system.

\newpage
\rule{\textwidth}{0.5mm}
{\bf Equation set 2: Post-Translational Modifications (PTMs)}\\
\rule{\textwidth}{0.5mm}

We assume that  there are $PTM_{num}$ types of PTMs. Currently, we
talked about four types of PTMs; acetylation, ubiquitylation,
phosphorylation, and methylation. 

\begin{equation}
\label{eq:ptm}
\frac{d[P_1^0]}{dt} = [mRNA_1] \times k_{tsln\_{P_1}} - [P_1] \times
k_{deg\_{P_1}} - \sum_1^{PTM_{num}} [P_1] \times k_{fPTM} 
               + \sum_1^{PTM_{num}} [P_1] \times k_{rPTM} 
\end{equation}

\begin{equation}
\label{eq:ptm-2}
\frac{d[P_1^1]}{dt} = [P_1^0] \times k_{fPTM} - [P_1^1] \times k_{deg\_PTM} - [P^1_1] \times k_{rPTM}
\end{equation}


In equation (\ref{eq:ptm}), the first term refers to the basic proteins
that are being created, the second term refers to the basic proteins
that are being degraded, the third term, which is a summation, refers to
the basic proteins that are being post-translationally modified in any
of the $PTM_{num}$ ways, and the fourth term refers to the
post-translationally modified proteins that are being reversed.

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{\bf Equation set 3: Protein-Protein Complexation}\\
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When two proteins $P_1$ and $P_2$ form a complex, the concentration of
both decrease. In the first two equations, the first term refers to the proteins
being transcribed from mRNA. The second term refers to the proteins being degraded.
The third term refers to the proteins forming a complex and the last
term refers to the complex being reversed. 

\begin{equation}
\label{eq:complex1}
\frac{d[P_1^0]}{dt} = [mRNA_1] \times k_{tsln\_{P_1}}
  - [P_1] \times k_{deg\_{P_1}} 
  - [P_1^0] \times  [P_2^0] \times k_{complex_1} 
  + [P_1^0 \, P_2^0] \times k_{r\_complex_1} 
\end{equation}

\begin{equation}
\label{eq:complex2}
\frac{d[P_2^0]}{dt} = [mRNA_2] \times k_{tsln\_{P_2}}
  - [P_2] \times k_{deg\_{P_2}} 
  - [P_1^0] \times  [P_2^0] \times k_{complex_1} 
  + [P_1^0 \, P_2^0] \times k_{r\_complex_1} 
\end{equation}

The below formula shows that the concentration of the complex increases
as it forms and decreases as the complex reverses or degrades.
\begin{equation}
\label{eq:complex3}
\frac{d[P_1^0 P_2^0]}{dt} = 
  [P_1^0] \times  [P_2^0] \times k_{complex_1} 
  - [P_1^0 \, P_2^0] \times k_{r\_complex_1} 
  - [P_1^0 \, P_2^0] \times k_{deg\_complex_1} 
\end{equation}

\newpage
\rule{\textwidth}{0.5mm}
{\bf Equation set 4: Protein-Gene Interactions ($P_1^0 \, P_2^0$
reciprocal regulation)}\\
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Reciprocal regulation means that two proteins bind to each other's
promoter and can act as repressors or activators (Type ii in Fig. 2 of
Hakim et al.'s paper). This binding changes the rate of production for
each protein involved. The proteins that are bound are not available in
the free pool as shown in the third term. The fourth term represents the
proteins that are becoming available due to unbinding.

\begin{equation}
\label{eq:recip-reg-0}
\frac{d[mRNA_1]}{dt} =
Rfac \times G_1 \times k_{txn_1}
  - [mRNA_1] \times k_{deg\_mRNA_1}
\end{equation}

\begin{equation}
\label{eq:recip-reg}
\frac{d[P_1^0]}{dt} = [mRNA_1] \times k_{tsln\_{P_1}}
  - [P_1] \times k_{deg\_{P_1}} 
  - [P_1^0] \times  (k_{bind\_{P_1}} - k_{unbind\_{P_1}})
\end{equation}


Note that when a protein binds to its own gene, the process is called
autoregulation. 

Next, we will put in the last two lines of the equations for reciprocal
regulation. 

\begin{equation}
\label{eq:recip-reg-2}
\frac{d[P_2^0]}{dt} = [mRNA_2] \times k_{tsln\_{P_2}}
  - [P_2] \times k_{deg\_{P_2}} 
  - [P_2^0] \times  k_{bind\_{P_2 G_1}}
  + [P_2^0(G_1)] \times k_{unbind\_{P_2 G_1}}
\end{equation}

The effect of this type of interaction is expressed as a regulation factor 
multiplied to the maximal rate of transcription.

\rule{\textwidth}{0.5mm}
{\bf Equation set 5: Mutation Factors (in progress)}\\
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To illustrate the common mutations, those that are most probable, we will walk through several example equations pointing out possible targets of the F (forward), D (degrade), R (reverse), and H(methyl), H(acetyl) (histone) kinetic modifiers/'mutations'.

\begin{equation}
\label{eq:mut-1}
\frac{d[mRNA_1]}{dt} =
H(acetyl) \times H(methyl) \times G_1 \times k_{txn_1} 
- [mRNA_1] \times D_{1} \times k_{deg\_mRNA_1}
\end {equation}

where predominant H(acetyl), histone acetylation which ranges from 1 to 2 (or a set maximum) will result in result in increased transcription.  Histone methylation, H(methyl), will act as a gene silencer expressed as a multiplying factor from 0 to 1.  In the case of any target genes higher levels of control over these factors will be implemented (ie. important genes will not be turned off, H(methyl)=0. 

D is a degradation multiplier and can affect any degradation term, and the choice of term will result randomly from existing interactions.  D directly alters the value of kdeg term.

\begin{equation}
\label{eq:mut-2}
\frac{d[P_1^0]}{dt} = [mRNA_1] \times F_{1} \times k_{tsln\_{P_1}} 
- [P_1^0] \times D_{2} \times k_{deg\_{P_1}}
\end {equation}

We see a second instance of D, which occurs as a separate event, here illustrating another potential degradation target, and a forward mutation acting to increase or decrease transcription of the basic protein.  To exhibit R, a reversal factor, the protein must be involved in a reversible reaction type including: post-translational modification, protein-protein interactions, and protein-promoter binding.

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